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Algebraic covers: Field of moduli versus field of definition. (English) Zbl 0906.12001
Let $$B$$ be a regular, projective, geometrically irreducible variety over a field $$K$$, let $$X\to B$$ be a finite cover of $$B$$, generically unramified, with $$X$$ normal. It is a priori defined over $$K^{\text{sep}}$$, the separable closure of $$K$$. Suppose that $$X\to B$$ is isomorphic to each of its conjugates by $$\text{Gal} (K^{\text{sep}}/K)$$. Then, it is natural to ask the following question: is $$X\to B$$ defined over $$K$$? The answer is no in general. A counter-example is known (Couveignes-Granboulan), moreover, it was known that, for $$G$$-covers, i.e. for Galois covers with group $$G$$, the obstruction is measured by one characteristic class in $$H^2(K,Z(G))$$ ($$Z(G)$$ is the center of $$G$$). The authors study the obstruction in the above context. They prove that the situation is more complicated: the obstruction is controlled by finitely many characteristic classes in $$H^2(K,Z(G))$$, where in that case $$G$$ is the group of the cover. The methods that they have used yield new concrete criteria to prove that a cover is defined over $$K$$. A particularly beautiful application is the following theorem (conjectured by E. Dew): a $$G$$-cover is defined over $$\mathbb{Q}$$ if and only if it is defined over $$\mathbb{Q}_p$$ for all places $$p$$ (including $$p=\infty$$).

##### MSC:
 12F12 Inverse Galois theory 14H30 Coverings of curves, fundamental group 11G35 Varieties over global fields 11G25 Varieties over finite and local fields
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